{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T00:25:09Z","timestamp":1720657509278},"reference-count":0,"publisher":"Combinatorial Press","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ars Comb."],"published-print":{"date-parts":[[2024,6,30]]},"abstract":"<jats:p>One of the fundamental properties of the hypercube \\( Q_n \\) is that it is bipancyclic as \\( Q_n \\) has a cycle of length \\( l \\) for every even integer \\( l \\) with \\( 4 \\leq l \\leq 2^n \\). We consider the following problem of generalizing this property: For a given integer \\( k \\) with \\( 3 \\leq k \\leq n \\), determine all integers \\( l \\) for which there exists an \\( l \\)-vertex, \\( k \\)-regular subgraph of \\( Q_n \\) that is both \\( k \\)-connected and bipancyclic. The solution to this problem is known for \\( k = 3 \\) and \\( k = 4 \\). In this paper, we solve the problem for \\( k = 5 \\). We prove that \\( Q_n \\) contains a \\( 5 \\)-regular subgraph on \\( l \\) vertices that is both \\( 5 \\)-connected and bipancyclic if and only if \\( l \\in \\{32, 48\\} \\) or \\( l \\) is an even integer satisfying \\( 52 \\leq l \\leq 2^n \\). For general \\( k \\), we establish that every \\( k \\)-regular subgraph of \\( Q_n \\) has \\( 2^k, 2^k + 2^{k-1} \\) or at least \\( 2^k + 2^{k-1} + 2^{k-3} \\) vertices.<\/jats:p>","DOI":"10.61091\/ars159-14","type":"journal-article","created":{"date-parts":[[2024,7,10]],"date-time":"2024-07-10T05:14:41Z","timestamp":1720588481000},"page":"165-178","source":"Crossref","is-referenced-by-count":0,"title":["5-Regular Subgraphs in Hypercubes"],"prefix":"10.61091","volume":"159","author":[{"given":"Y. M.","family":"Borse","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"name":"Department of Mathematics, Savitribai Phule Pune University, Ganeshkhind, Pune 411007, India","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S. R.","family":"Shaikh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J. B.","family":"Saraf","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"name":"Department of Mathematics, Savitribai Phule Pune University, Ganeshkhind, Pune 411007, India","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"name":"Department of Mathematics, Savitribai Phule Pune University, Ganeshkhind, Pune 411007, India","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"39747","published-online":{"date-parts":[[2024,6,30]]},"container-title":["Ars Combinatoria"],"original-title":[],"deposited":{"date-parts":[[2024,7,10]],"date-time":"2024-07-10T05:14:55Z","timestamp":1720588495000},"score":1,"resource":{"primary":{"URL":"https:\/\/combinatorialpress.com\/ars-articles\/volume-159\/5-regular-subgraphs-in-hypercubes\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,30]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,6,30]]},"published-print":{"date-parts":[[2024,6,30]]}},"URL":"https:\/\/doi.org\/10.61091\/ars159-14","relation":{},"ISSN":["0381-7032","2817-5204"],"issn-type":[{"value":"0381-7032","type":"print"},{"value":"2817-5204","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,30]]}}}